From chaos to calm

Nonlinear mapping tends to maintain nonlinearity the farther its trajectory from a fixed point. Is there an example of a trajectory reverting to linearity after traversing a nonlinear region? I am especially interested in local, linear special relativity transitioning to global, nonlinear general relativity, and a possible converse.

Yes. Any object entering the Solar System from outside came from a linear trajectory, experiences a hyperbolic trajectory in the vicinity of the Sun and reverts to linear after it has left the Solar System.

Does general relativity behave nonlinearly, yet not chaotically? I believe what I seek is a near-linear gravitational trajectory becoming a more chaotic one, then becoming near-linear again. Sorry for the confusion.

I'm still not quite sure what you're looking for but here are a couple of ideas that I think resemble what you seem to be asking.

Some aspects of weather go from calm (normal weather) to chaotic behavior (hurricanes) and back again to calm.

Plasma waves can produce an "echo" meaning that when such a wave is launched it is well structured but as it propagates away it is damped (Landau damping). The wave disappears but later downstream the wave re-emerges. This process can repeat several times.

Regarding general relativity, the motion of objects (including light) close to a massive object can be chaotic but I am not aware of a situation where an object that enters close to the object (event horizon, e.g.) completely escapes.

Hello all.
Just a few observations.
Our solar sisem is chaotic by nature (large time scail). Our life span is just too short for us to notice.
Also weather is chaotic by nature, that means that in calm weather and in mid huricane it is impossible to predict future events for distant future, even with lots of acure data (not equaly unpredictable, but still unpredistable).

@LorenBoda, So you are looking for transitional chaos in damped systems. I think if you want to include general relativity you'll have to be more specific, but there is an example of a transitional chaos involving gravity.

It is an experiment with three magnets (positioned at the corners of a tiangle) and an iron ball that hangs above them exactly at the center.

When you release the ball its movement is linear (it's a damped pendulum). When the amplitude weakens it becomes chaotic (it can't decide which magnet attracts it more). When amplutide weakens even more, it settles and osscilates arround one of the magnets (yt the beggining you can't tell which one).

LQG: The main difference that I see is that Nottale's approach indeed modifies General relativity, while Schleich and Witt are only giving a new definition of chaos, maintaining the habitual GR structure. For example, you must know that in GR spacetime is a differentiable manifold. But looking this Nottale's paper http://arxiv.org/abs/hep-th/0307093
he says there that in his theory spacetime is a non-differentiable manifold

Take a strong (like an NdFeB or SmCo5) magnet in the shape of say a cylindrical disc of diameter D, magnetized along the direction of the axis. Set this on your table. Now over this slide a transparent (so it's nicer to see) plastic tube/pipe of inner diameter just barely greater than D, but have it be longer than the height (or thickness) or the magnet. Now take another magnet, identical to the first and drop it down the pipe. If you dropped it with opposite poles facing each other, this second magnet will levitate above the fist one.

I can't easily picture a geometry between only ferromagnets that will give rise to a stable equilibrium - but maybe I'm just not thinking right. In the above example, there is a point of equilibrium (where the magnet can levitate without any mechanical contact whatsoever), but it is unstable - if slightly disturbed, the magnet will flip and fall into the other one.

In the case of a reasonable diamagnet, it is actually possible to achieve levitation at the highest achievable fields today. With a reasonably high field magnet, it is possible to stably levitate say, a drop of water or a piece of bismuth - though I think that the better approach is to levitate the magnet between bismuth plates.

Someone actually levitated a frog with a pulsed field magnet. A google search might prove fruitful.